Abstract. We discuss a general framework for the numerical solution of a family of semilinear elliptic problems whose leading differential operator is the Laplacian. A problem is first transformed to one on a standard domain via a conformal mapping. The boundary value problem on the standard domain is then reduced to an equivalent integral operator equation. We employ the Galerkin method to sol...

 In this paper, we present "a posteriori" analysis of the fundamental concepts involved in the modelling of problems of mathematical physics by Partial Differential Equations (PDEs). Our aim is to improve our students' understanding of PDEs when applied to an engineering problem, from a completely qualitative point of view. They should be able to understand the deep meaning of any Laplacian, c...

In this paper, we consider a nonhomogeneous space-time fractional telegraph equation defined in a bounded space domain, which is obtained from the standard telegraph equation by replacing the firstor second-order time derivative by the Caputo fractional derivative Dt , α > 0; and the Laplacian operator by the fractional Laplacian (−∆) , β ∈ (0, 2]. We discuss and derive the analytical solutions...

Content The first paper by Jean Mawhin on critical point theory [] was published in  and was devoted to periodic solutions of a forced pendulum equation. One of the most recent papers in  [] concerns periodic solutions of difference systems with φ-Laplacian. It is impossible to describe all the contributions. We have selected  articles,  books and some fundamental topics: the force...

Let Ω be a bounded domain in R . In this paper, we consider the following nonlinear elliptic equation of N -Laplacian type: (0.1) { −∆Nu = f (x, u) u ∈ W 1,2 0 (Ω) \ {0} when f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosett...

In this paper we study the heat equation (of Hodge Laplacian) deformation of .p; p/-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a .p; p/-form solution is preserved under such an invariant condition, we prove the sharp differential Harnack (in the sense of LiYau-Hamilton) estimates for the positive solutions of the Hodge Laplacian heat equa...

Given a Carnot-Carathéodory metric space (R, dcc) generated by vector fields {Xi} m i=1 satisfying Hörmander’s condition, we prove in theorem A that any absolute minimizer u ∈ W 1,∞ cc (Ω) to F (v,Ω) = supx∈Ω f(x,Xv(x)) is a viscosity solution to the Aronsson equation (1.6), under suitable conditions on f . In particular, any AMLE is a viscosity solution to the subelliptic ∞-Laplacian equation ...

In this paper we give the definition of 1/2-harmonic maps u : D → N , where D is an open set of IR and N is a k dimensional submanifold N of IRm as critical points of the functional ∫ Ω |∆ 1/4u|2dx . We write the Euler-Lagrange equation satisfied in a weak sense by the 1/2-harmonic maps, which is a nonlocal partial differential equation involving the 1/2-Laplacian operator . We prove the local ...

has this character. Even obvious results for this equation may require advanced estimates in the proofs. We refer to the books [DB] and [WZYL] about this equation, which is called the “evolutionary p-Laplacian equation,” the “p-parabolic equation” or even the “non-Newtonian equation of filtration.”. Our objective is to study the regularity of the viscosity supersolutions and their spatial gradi...

wt t (t, x)− K (‖wx (t, ·)‖ β Lr (R))a(wx (t, x))wxx (t, x) = 0, (2) w(0, x) = 8(x), wt (0, x) = 9(x), where K is an arbitrary function, sufficiently smooth and taking only positive values; and a = a(s) behaves like |s|p−2 near s = 0. The detailed assumptions on K , r , β, and a are given in (3), (4) and Condition 1 below. For K = K (s) = c1 + c2s (c1, c2 > 0) and p = r = β = 2, we get the famo...